Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices

TL;DR

This paper analyzes the asymptotic behavior of perfect matchings on large contracting square-hexagon lattices, establishing limit shape, height fluctuations, and connections to random matrix theory.

Contribution

It provides explicit formulas for the partition function via Schur functions and proves limit shape and Gaussian free field convergence for the height functions.

Findings

Derivation of explicit partition function formulas using Schur functions.

Proof of the law of large numbers and Gaussian fluctuations for height functions.

Identification of the frozen boundary as a cloud curve with tangent points depending on lattice parameters.

Abstract

We study asymptotics of perfect matchings on a large class of graphs called the contracting square-hexagon lattice, which is constructed row by row from either a row of a square grid or a row of a hexagonal lattice. We assign the graph periodic edge weights with period $1\times n$, and consider the probability measure of perfect matchings in which the probability of each configuration is proportional to the product of edge weights. We show that the partition function of perfect matchings on such a graph can be computed explicitly by a Schur function depending on the edge weights. By analyzing the asymptotics of the Schur function, we then prove the Law of Large Numbers (limit shape) and the Central Limit Theorem (convergence to the Gaussian free field) for the corresponding height functions. We also show that the distribution of certain type of dimers near the turning corner is the same as the eigenvalues of Gaussian Unitary Ensemble, and that in the scaling limit under the boundary condition that each segment of the bottom boundary grows linearly with respect the dimension of the graph, the frozen boundary is a cloud curve whose number of tangent points to the bottom boundary of the domain depends on the size of the period, as well as the number of segments along the bottom boundary.